Angles with Parallel LinesAngles with Parallel Lines 
                  Topic Index | Geometry Index | Regents Exam Prep Center




      A transversal is a line that intersects two or more lines (in the same 
      plane).  When lines intersect, angles are formed in several locations.  
      Certain angles are given "names" that describe "where" the angles are 
      located in relation to the lines.  These names describe angles whether the 
      lines involved are parallel or not parallel.
      Remember that:
             - the word INTERIOR means BETWEEN the lines.
             - the word EXTERIOR means OUTSIDE the lines.
             - the word ALTERNATE means "alternating sides" of the transversal. 
            When the lines are NOT parallel ...               When the lines are 
            parallel...


       
            The names "alternate interior angles", "alternate exterior angles", 
            "corresponding angles", and "interior angles on the same side of the 
            transversal" are used to describe specific angles formed when lines 
            intersect.  These names are used both when lines are parallel and 
            when lines are not parallel.
            Let's examine these angles, and other angles, when the lines are 
            parallel.

       
                  When the lines are parallel:
                  Alternate Interior Angles
                  (measures are equal)
                  The name clearly describes "where" 
                  these angles are located.  

             
                  Look carefully at the diagram below:


      Hint:  If you draw a Z on the diagram, the alternate interior angles
      are found in the corners of the Z.  The Z may also be a backward Z.
                 
                        Theorem: 
                  If two parallel lines are cut by a transversal, the alternate 
                  interior angles are congruent.


       
                        Theorem: 
                  If two lines are cut by a transversal and the alternate 
                  interior angles are congruent, the lines are parallel.



       
                  When the lines are parallel:
                  Alternate Exterior Angles
                  (measures are equal)
                  The name clearly describes "where" 
                  these angles are located. 


                  Look carefully at the diagram below: 


       
                        Theorem: 
                  If two parallel lines are cut by a transversal, the alternate 
                  exterior angles are congruent.


       
                        Theorem: 
                  If two lines are cut by a transversal and the alternate 
                  exterior angles are congruent, the lines are parallel.




       
                  When the lines are parallel:
                  Corresponding Angles
                  (measures are equal)
                  Unfortunately, the name of these angles
                  does not clearly indicate "where" they
                  are located.  They are located:
                  - on the SAME SIDE of the transversal
                  - one INTERIOR and one EXTERIOR
                  - and they are NOT adjacent (they don't touch).
                  (They lie on the same side of the transversal,
                  in corresponding positions.)



      Hint:  If you took a picture of one corresponding angle and slid 
      the angle up (or down) the same side of the transversal, you
      would arrive at the other corresponding angle. 
      Also:  If you draw an F on the diagram, the corresponding angles can be 
      found in the "corners" of the F.  The F may be backward and/or 
      upside-down. 
            
       
                        Theorem: 
                  If two parallel lines are cut by a transversal, the 
                  corresponding angles are congruent.


       
                        Theorem: 
                  If two lines are cut by a transversal and the corresponding 
                  angles are congruent, the lines are parallel.


       
       
                  When the lines are parallel:
                  Interior Angles on the Same Side of the Transversal
                  (measures are supplementary)
                  Their "name" is simply a description of where the angles are 
                  located.
                   



       
                        Theorem: 
                  If two parallel lines are cut by a transversal, the interior 
                  angles on the same side of the transversal are supplementary.


       
                        Theorem: 
                  If two lines are cut by a transversal and the interior angles 
                  on the same side of the transversal are supplementary, the 
                  lines are parallel.


       
       

      Of course, there are also other angle relationships occurring when working 
      with parallel lines.  
                  Vertical Angles
                  (measures are equal)
                  Vertical angles are ALWAYS equal, whether you
                  have parallel lines or not.


               Refresh your memory using the diagram below:


                        Theorem: 
                  Vertical angles are congruent.


       
       
                  Angles forming a Linear Pair 
                  (Adjacent Angles creating a Straight Line)
                  (measures are supplementary)

                  This is an "old" idea about angles revisited.  Since a
                 straight angle contains 180, these two adjacent angles
                 add to 180.  They form a linear pair.
               (Adjacent angles share a vertex, share a side, and do not 
            overlap.)


                        Theorem: 
                  If two angles form a linear pair, they are supplementary.


       
       



                  Topic Index | Geometry Index | Regents Exam Prep Center 
                  Created by Donna Roberts
                  Copyright 1998-2011 http://regentsprep.org 
                  Oswego City School District Regents Exam Prep Center 
                

